On types of elements, Gelfand and strongly harmonic rings of skew PBW extensions over weak compatible rings

TL;DR

This paper characterizes various algebraic elements and explores Gelfand and harmonic ring properties in skew PBW extensions over weak compatible rings, extending known results from polynomial rings to broader noncommutative contexts.

Contribution

It introduces new characterizations of elements and ring properties in skew PBW extensions over weak compatible rings, expanding the understanding beyond classical polynomial rings.

Findings

Characterization of units, idempotents, and regular elements in skew PBW extensions.

Extension of Gelfand and harmonic ring concepts to noncommutative skew PBW extensions.

Generalization of polynomial ring results to broader algebraic structures.

Abstract

We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $\pi$-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.